Heyting, Lindenbaum and Multiple Conclusions

One of the odd things about my preferred logic of vagueness, as set out in True, Truer, Truest is that disjunctions can be perfectly true even though no disjunct is perfectly true. Some people (including me some of the time) find this a little unintuitive. It also has a quirky logical consequence.

I hadn’t been thinking about this at the time, but one consequence of that is that the simplest way to do multiple conclusion logic in the theory will lead to some odd failures of validity. For instance, using ‘:’ as a turnstile, it is natural to think that (1) will fail.

(1) A ∨ B: A, B

If our understanding of validity is that on any interpretation, at least one of the conclusions is at least as true as the conjunction of the premises, then (1) is false. That’s because when B is, say, ¬A, then the premise will be perfectly true, but each disjunct will be less than perfectly true.

Objection: It’s a little odd to have this definition of validity. It requires, in effect, that we treat the premises as a long conjunction, but the conclusions separately. If the rule for validity was that the disjunction of the conclusions must be at least as true as the conjunction of the premises, then (1) would be valid.

Response: I think this objection ultimately doesn’t take the multiple conclusion form of (1) seriously enough. It’s true that you could turn around and say that the theory of validity in True, Truer, Truest doesn’t take the multiple premise form of arguments seriously enough. But to that I happily plead guilty; I’m happy to say that arguments really have one long premise, with the parts sometimes joined by ‘∧’, and sometimes by ‘.’.

I mentioned briefly in the earlier paper that you could do an intuitionist version of the semantics in True, Truer, Truest. The classical version that I developed there said that truer generated a Boolean lattice over (equivalence classes of) propositions (where the classes are of equally true propositions). If instead you said that truer generates a Heyting lattice over (equivalence classes of) propositions, you get an intuitionist version of my theory. (That is, all and only intuitionist theorems are logical truths, and and all and only intuitionistically valid sequents are really valid.)

Now some Heyting lattices have the property that the join of a and b can be the top element even though neither a nor b is the top element. But not all Heyting lattices have this property. In particular, the Lindenbaum-Tarski algebra has this property. That’s the algebra defined over sentences in a countable language ordered by intuitionist entailment. So if we supplement the theory of True, Truer, Truest with the requirement that A ∨ B is perfectly true iff A is perfectly true or B is perfectly true, we get a consistent intuitionist theory.

Having said that, we still don’t make (1) valid. It can still be the case that A ∨ B is strictly weaker than both A and B, and in that case, (1) will be invalid. The trick, I think, will be to change the definition of validity.

New Account of Validity: An argument is valid iff whenever the premises are perfectly true, at least one of the conclusions is perfectly true.

The New Account of Validity doesn’t change what sequents are valid in the classical case, but it does change some validities in the (multiple-conclusion) intuitionist version of the ‘truer’ theory. Note that the New Account doesn’t imply that p entails that p is perfectly true, because p can be perfectly true without it being perfectly true that p is perfectly true.

If we think the algebra of truth values is a Lindenbaum-Tarski algebra and we adopt the New Account of Validity, then (1) will be valid. I’m not sure whether this is too high a price to pay to keep (1) valid, but I think there is some motivation for each of these assumptions.

UPDATE: I fixed the broken link to True, Truer, Truest. Thanks to Seamus Bradley for alerting me to this.

3 Replies to “Heyting, Lindenbaum and Multiple Conclusions”

I’m finding it a bit hard to see what turns on this. The question whether (1) holds on its own doesn’t seem to be substantive unless I’ve got some independent grasp of “:”. (Which I personally don’t unless you have some more specific reading of the multiple conclusion entailment relation in mind. I, for example, know when “X:Y” holds in the technical setting where it’s defined, but don’t have intuitions about whether (1) holds in this unfamiliar context.)

Maybe it would help if you could say something about the normative role “:” is supposed to play, e.g., some restriction that it imposes on rational credences, or something… but that’s not just the trivial fact that you can accept (p or ~p) without accepting p and without accepting ~p.

You’re right Andrew, trading intuitions about entailment in multiple conclusion logics is not really a way forward, and this probably wasn’t a useful framing device.

Here’s what I was thinking more clearly.

It can seem very odd that a theory can allow for a disjunction to be perfectly true, although neither disjunct is perfectly true. Supervaluational theories have this property, and so does my degrees of truth theory.

One consequence of this odd feature of supervaluational/degree of truth theories is that the invalidate sequents like (1).

So, rather than talk directly about truth, we might look at whether (1) holds in various theories as a means for getting at whether the theory really avoids this odd consequence.

But I’m not sure this little gambit worked. It would probably have been better to stick to focussing on truth conditions for disjunctions, and not start mucking about with multiple conclusion logics.

One way of making the intuition stronger for me (as, again, I don’t really have good intuitions about these unfamiliar non-disquotational notions of truth) is to say that, when a MC entailment holds, it’s not rationally permissible to accept all the of premisses and reject one of the conclusions. If that’s the upshot of the failure of (1) then it’s clear to me that we’re not just disagreeing about how to define a piece of jargon.

I think you could probably make a good case that your original definition of “:” doesn’t have this consequence (should we reject things that almost but not perfectly true?) Although I think Field, when he was a classical guy, has made the failure of (1), understood this way, into a neat theory of vagueness that clearly distinguishes it from ignorance accounts.